Rank of Unit Group of Burnside Ring: Learning Group Action and Orbit & Stabilizer Theorem

This research project aims to automate the process of calculating the rank of the unit group of Burnside ring in GAP. After learning to use GAP during the first week, the new goal is to calculate the rank by hand. Based on a lot of literature review and talking to Dr. Carman, I wanted to first approach this problem by using the idea of table of marks, which requires profound understanding of group action, Burnside’s lemma, and isomorphism class, which were not covered in the abstract algebra class I took. I spent my second week learning about the concepts by both searching online and asking Dr. Carman during our meeting time. I mainly studied group action and orbit & stabilizer theorem, I later typed the essential statements in LaTex. ( See graphs below)

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Rank of Unit Group of Burnside Ring: Learning GAP

    The research objective is to find the mathematical theory of the size of the unit groups of Burnside rings. Specifically, it focuses on automating the algorithms of calculating the size of unit group of Burnside ring by employing GAP (Groups, Algorithms, Programming). After the first meeting with Dr. Carman, I wrote several things on my to-do list for the first week of research. As a large amount of the computational work of this research project will require the assistance of GAP, the main goal of the first week is to learn to use GAP by gathering relevant materials and reading tutorials and documents.

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Kernels

I finally got the model working! The kernel looks good and its graph is pretty (below). I’ve used some iterative code to find the model’s stable population growth rate, stable stage distribution, and reproductive value vector. I have not used a very large matrix due to the computing limitations of my laptop, but I can easily transfer my code to a more powerful computer when it becomes necessary.

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Some Direction

I was a little hazy on what I’d actually be doing this summer. Would I be helping someone else with her research? Would I be using data that was already collected? Or would I be messing around until I had the opportunity to head to Maine and Vermont to collect new data? It turns out I’m doing a little bit of each. There is an incredible amount of data being used for a matrix model that has been in the works longer than I’ve been at W&M, and I am helping sort through it, picking out funky pieces of data and marking them so they can be checked with the real trees when we take the trip to Maine. I am also using the current data to practice my modeling skills. One goal for the summer is having a functional model for the Maine data. I am using a statistical program, R, that is very different than Python ( the language I’ve used so far in my CS classes), and I spend a significant amount of time just trying to find proper syntax and functions for the things I want to do. Once more data is collected, there will be two years of data collected from the Vermont site, and I spend time this fall building a new model for a new set of data.

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